Hello :)
I've been a DT770 user for almost 2 years
Hey,
Its a pretty easy fix, but you need certain tools and previous experience helps..
1st you need to purchase a new stereo plug for the headphones.
It won't cost you much.
Your going to need a soldering iron, solder, and something to cut the wires with.
If you have a soldering iron, then you can solder the wires to the wires that come from the new plug.
Soldering the wires is probably the best solution, another way would be to use insulated terminals to connect the wires, you can buy a set from a hardware store. It is probably the easier option if you don't own a soldering iron. but it wont be as secure as soldering.
The soldering is simple, the most difficult part is to find out which wire goes where. There will be at least 3 wires, Ground, left and right. If you get one or two wires wrong, you might have no sound at all, no sound from other speaker, or the right speakers sound might come from the left and so on.
as for finding out which wire is what, you would need to use a multimeter to test which wire goes where, and the one that goes to both would be ground.
If you don't own a iron, you have never soldered, and you don't want to try, you might be best off getting someone else to do it for you, replacing a stereo plug, shouldn't cost too much at a general electronics repairer or a handy friend might do it for free.
Heres a simple good video on how to solder.